Actually, two-way thermal-energy exchange between working fluid and solid material of a casing is a leading problem for modern - semi automatic - design techniques. Many questions should be solved, especially, the turbulent mode of thermal energy transport both in fluid and solid, should be re-examined and reformulated from the primary principles. In the present paper, a group of researchers from Energy Conversion Department of IMP PAN at Gdańsk, tries to summarise a last three-years efforts towards to mathematical modelling of advanced models of thermal energy transport. This extremely difficult problem in "thermal-FSI" ("Fluid Solid Interaction") means that the both for solid and fluid mathematical model of a surface layer should be self-equilibrated and self-concise. Taking these requirements into account, an advanced Reynolds-Stanton analogy has been discussed and implemented. Some numerical examples concerning of the benchmarks experiments and industrial applications have also been developed and presented.